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To simplify the given complex expression:

\[
\frac{24 + 3i}{6 - 3i}
\]

we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(6 - 3i\) is \(6 + 3i\).

### Step-by-step solution:

1. **Multiply the numerator and denominator by the conjugate of the denominator**:
   \[
   \frac{24 + 3i}{6 - 3i} \times \frac{6 + 3i}{6 + 3i} = \frac{(24 + 3i)(6 + 3i)}{(6 - 3i)(6 + 3i)}
   \]

2. **Expand the numerator** using distributive property (FOIL):
   \[
   (24 + 3i)(6 + 3i) = 24 \times 6 + 24 \times 3i + 3i \times 6 + 3i \times 3i
   \]
   Simplifying:
   \[
   24 \times 6 = 144, \quad 24 \times 3i = 72i, \quad 3i \times 6 = 18i, \quad 3i \times 3i = 9i^2
   \]
   Since \(i^2 = -1\), the expression becomes:
   \[
   144 + 72i + 18i - 9 = 135 + 90i
   \]

3. **Simplify the denominator** using the difference of squares formula:
   \[
   (6 - 3i)(6 + 3i) = 6^2 - (3i)^2 = 36 - 9i^2
   \]
   Since \(i^2 = -1\), this becomes:
   \[
   36 - 9(-1) = 36 + 9 = 45
   \]

4. **Combine the results**:
   \[
   \frac{135 + 90i}{45}
   \]

5. **Simplify by dividing both terms by 45**:
   \[
   \frac{135}{45} + \frac{90i}{45} = 3 + 2i
   \]

Thus, the expression in the form \(a + bi\) is:

\[
3 + 2i
\]

Would you like further details or explanations on any step?

### Related questions:
1. What is the importance of using the conjugate in complex division?
2. How do you determine the conjugate of a complex number?
3. Why is \(i^2\) equal to \(-1\)?
4. How can you multiply two complex numbers using the distributive property?
5. What other methods can be used to simplify complex expressions?

### Tip:
When simplifying complex fractions, always multiply by the conjugate of the denominator to eliminate the imaginary part from the denominator.

Express the complex number (24 + 3i) / (6 - 3i) in simplest a + bi form.

Learn how to simplify the complex number (24 + 3i) / (6 - 3i) using the conjugate method. This step-by-step solution includes key concepts such as complex number multiplication, distributive property, and eliminating the imaginary part from the denominator. Suitable for Grades 11-12.

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